a: \(A=\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}=\left(\dfrac{2}{x_1-2}-\dfrac{2}{x_2-2}\right)\cdot\dfrac{1}{x_1-x_2}\)
\(=\dfrac{2x_2-4-2x_1+4}{\left(x_1-2\right)\left(x_2-2\right)}\cdot\dfrac{1}{x_1-x_2}=\dfrac{-2}{\left(x_1-2\right)\left(x_2-2\right)}\)
TH1: x1>2; x2>2
=>(x1-2)(x2-2)>0
=>A<0
=>Hàm số nghịch biến
TH2: x2<2; x1<2
=>(x1-2)(x2-2)>0
=>A<0
=>hàm số nghịch biến
b: \(B=\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}\)
\(=\left(\dfrac{x_1+1}{x_1-2}-\dfrac{x_2+1}{x_2-2}\right)\cdot\dfrac{1}{x_1-x_2}\)
\(=\dfrac{x_1x_2-2x_1+x_2-2-x_1x_2+2x_2-x_1+2}{\left(x_1-2\right)\left(x_2-2\right)}\cdot\dfrac{1}{x_1-x_2}\)
\(=\dfrac{-3}{\left(x_1-2\right)\left(x_2-2\right)}\)
TH1: x1>2; x2>2
=>(x1-2)(x2-2)>0
=>B<0
=>Hàm số nghịch biến
TH2: x2<2; x1<2
=>(x1-2)(x2-2)>0
=>B<0
